Lorenz Formulas: T-zero & T Explained

In summary: Thanks for the explanation, Dodo.It's a mystery to me why "spacetime diagrams" are... interesting. In summary, In the time lorenz formulas, how can u tell what the difference between t-zero and T is? t-zero T= -------------------- ------------------- | 1 - (v^2/c^2
  • #1
Moe_the_Genius
13
0
In the time lorenz formulas, how can u tell what the difference between t-zero and T is?
t-zero
T= --------------------
-------------------
| 1 - (v^2/c^2)
\|

for example in the following problem:
With what speed will a clock have to be moving in order to run at rate that is one-half the rate of a clock at rest?

How can u know what to plug in for t-zero and T?

Thanks so much
 
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  • #2
Moe_the_Genius said:
In the time lorenz formulas, how can u tell what the difference between t-zero and T is?
t-zero
T= --------------------
-------------------
| 1 - (v^2/c^2)
\|

for example in the following problem:
With what speed will a clock have to be moving in order to run at rate that is one-half the rate of a clock at rest?

How can u know what to plug in for t-zero and T?

Thanks so much
I have always found it very helpful in these cases for follow the derivation of the formula. Then you'll understand it much better. See - http://www.geocities.com/physics_world/sr/light_clock.htm

Pete
 
  • #3
Hint: your T and t-zero are not points in time, but time intervals.
 
  • #4
thank u for the tips yet still I get the wrong number when I plug my numbers in:

-->With what speed will a clock have to be moving in order to run at rate that is one-half the rate of a clock at rest?

For the problem, I plug in T=(1/2)t and t-zero=t, because I think the moving clock the time is one half-t, while to the clock at rest the time is t. Doesn't t-0 represent what the clock at rest observes??

thanks for any further help anyone provides
 
  • #5
Moe_the_Genius said:
For the problem, I plug in T=(1/2)t and t-zero=t, because I think the moving clock the time is one half-t, while to the clock at rest the time is t. Doesn't t-0 represent what the clock at rest observes??
[itex]T_0[/itex] represents the time as measured by the "moving" clock; [itex]T[/itex] represents the time as measured by "stationary" clocks.

[tex]T = \frac{T_0}{\sqrt{1 - v^2/c^2}}[/tex]

In your example, if the moving clock measures t seconds, then the stationary observers will measure 2t seconds according to their clocks. (They measure the moving clock to be running at half the normal rate.)
 
  • #6
This mysterious and often confusing formula of time-dliation can interpreted geometrically.
As Dodo says, "Hint: your T and t-zero are not points in time, but time intervals."


Let's introduce events:
the common meeting event O,
the distant event t0 on the moving clock [when the moving clock reads t0],
the local event T [when the stationary clock reads T], which our stationary observer says is simultaneous with the distant event t0. (Note that the moving observer does not regard these two events as simultaneous!)

OT and Ot0 are [timelike] legs of a [Minkowski]-right triangle on a spacetime diagram (as drawn below... time runs upwards by convention).

The legs of the triangle are OT, Ot0 and Tt0.
Ot0 is the "hypotenuse" of this [Minkowski]-right triangle.
OT is the "adjacent side" and Tt0 is the "opposite side"... these legs are [Minkowski]-perpendicular.
Another way to describe this is that the vector Ot0 is being resolved into temporal and spatial components by our stationary observer.
[By the way, the "angle" [tex]\theta [/tex] is called the rapidity... and it is related to the relative velocity by [tex]v=c\tanh\theta[/tex]. ]

So, think this way:

(adjacent side OT) = "COSINE(ANGLE)" (hypotenuse Ot0),
where "COSINE(ANGLE)" in the Minkowski geometry is [tex]\cosh\theta=\frac{1}{\sqrt{1-\tanh^2\theta}}=\frac{1}{\sqrt{1-(v/c)^2}}=\gamma[/tex]

[tex]\begin{picture}(200,200)(0,0)
\put(50,30){O}
\put(50,50){\textcolor{red}{\line(3,4){52}}}
\put(50,50){\textcolor{green}{\line(0,1){70}}}
\put(50,120){\textcolor{green}{\line(1,0){50}}}
\put(50,125){T}
\put(100,125){\[t_0\]}
\put(48,65){\[\theta\]}
\end{picture}
[/tex]

A spacetime diagram is worth a thousand words.
 
  • #7
robphy said:
A spacetime diagram is worth a thousand words.
That is quite true Rob. Someone asked a question here a while back which was best answered with a spacetime diagram. Here is the answer I gave to the question given at the top of the page

http://www.geocities.com/physics_world/sr/st_diagram.htm

I think the questioner was confusing spatial distances with spacetime intervals.

Pete
 
  • #8
It's a mystery to me why "spacetime diagrams" are rarely found in the relativity section of introductory textbooks... but they're everywhere in the [Galilean] kinematics section! They really can clear up a lot of ambiguities and misunderstandings.
 
  • #9
robphy said:
It's a mystery to me why "spacetime diagrams" are rarely found in the relativity section of introductory textbooks... but they're everywhere in the [Galilean] kinematics section! They really can clear up a lot of ambiguities and misunderstandings.
Good question. Let me see what I can learn on that. It would seem like a natural step since newer texts are introducing the spacetime interval and the spacetime diagram is awesome for describing that.

Pete
 
Last edited:

What are Lorenz Formulas?

Lorenz Formulas are mathematical equations developed by Edward Lorenz in the 1960s that describe the relationship between atmospheric variables such as temperature, wind speed, and humidity.

What is T-zero in Lorenz Formulas?

T-zero is the starting point for the time variable in Lorenz Formulas. It represents the initial condition of the atmosphere at a specific time.

What is T in Lorenz Formulas?

T in Lorenz Formulas represents time and is the variable being analyzed. It is used to track the changes in atmospheric variables over time.

How do Lorenz Formulas explain chaotic behavior in the atmosphere?

Lorenz Formulas demonstrate how small changes in initial conditions can lead to significant differences in future states of the atmosphere. This is known as the butterfly effect and is a key aspect of chaos theory.

What practical applications do Lorenz Formulas have?

Lorenz Formulas are used in weather forecasting, climate modeling, and other areas of atmospheric science to predict future states of the atmosphere. They also have applications in other fields, such as economics and biology, to model complex systems with chaotic behavior.

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